Problem: Isabella is making a huge flag of her country, the Republic of Seychelles, on a canvas $20$ by $10$ meters long. To do that, she has to draw a diagonal line that begins at the bottom-left corner and ends at the top side of the flag, $6.6$ meters to the right. Since the ends of the diagonal line are too far to put a ruler between them, Isabella wanted to find the angle of the diagonal and draw it using a protractor. What is the angle, in degrees, between the diagonal line and the left edge of the flag? Round your final answer to the nearest tenth.
The strategy Model the situation as a right triangle. Determine the appropriate trigonometric ratio in order to find the missing angle. Form an equation and solve for the missing angle. Calculate the final result and round. Modeling as a right triangle This situation can be modeled by the following right triangle. The height is $10\text{ m}$ and the base is $6.6\text{ m}$. We are asked to find the angle between the diagonal line and the left edge of the flag, which is the angle at the bottom. $?$ $10$ $6.6$ Determining the appropriate trigonometric ratio We are given the side ${\text{opposite}}$ to the missing angle and the side ${\text{adjacent}}$ to the missing angle. The appropriate trigonometric ratio is therefore the $\text{tangent}$. Forming an equation and solving Denoting the missing angle by $\theta$, we obtain the equation $\tan(\theta)=\dfrac{6.6}{10}$. Solving the equation, we get $\theta=\tan^{-1}\left(\dfrac{6.6}{10}\right)$. Evaluating this result in the calculator and rounding to the nearest tenth, we get $\theta=33.4^\circ$. Summary The angle between the diagonal line and the left edge of the flag is $33.4^\circ$.